A Simpler Analysis of the Bansal-Jiang Quasi Monte-Carlo Algorithm via Haar Wavelets

Abstract

Numerical integration---approximating the integral of a function $f$ using $n$ point evaluations---is a central task in science and engineering. The two main paradigms for this problem, the Monte Carlo and quasi-Monte Carlo methods, have distinct strengths and limitations, and a fundamental question is to design a method that combines the benefits of both. \smallskip Building on recent algorithmic advances in discrepancy theory, Bansal and Jiang \cite{BJ25a} gave a randomized QMC method that naturally bridges the MC and QMC error guarantees. Their method also achieves a surprising improvement over the classical Koksma--Hlawka inequality for QMC methods: it attains an error bound of $\tilde{O}({\sigma }_{\mathsf{SO}}(f)/n)$, where ${\sigma }_{\mathsf{SO}}(f)$ is a new notion of \emph{smoothed-out variation} that they introduced and showed to be substantially smaller than the Hardy--Krause variation governing the classical bound. \smallskip However, the analysis in \cite{BJ25a} is quite involved: it must carefully exploit the structure of the dyadic decomposition and the randomness of the algorithm inside a sufficiently fine discretization of the Hlawka--Zaremba formula to obtain cancellations among the high-frequency components in the Fourier decomposition of $f$. The contribution of this article is twofold: (1) We give an equivalent characterization of ${\sigma }_{\mathsf{SO}}(f)$ in terms of the Haar--Besov seminorm of $f$, relating this new notion of smoothed-out variation to classical quantities. (2) Through this characterization, we provide a conceptually simpler and more direct analysis of the Bansal--Jiang QMC method via Haar decomposition, bypassing the use of the Hlawka--Zaremba formula, Fourier decomposition, and the delicate cancellation arguments of \cite{BJ25a} that heavily exploit the structure of dyadic decomposition.